Summary and takeaways from Bridge Alternatives’ Portfolio Intuition (Link)
Assume:
Choose between A1 and A2
A1 | A2 | |
Return | 4.00% | 4.00% |
Volatility | 7.96% | 46.04% |
Correlation | -.20 | -.20 |
Unsurprisingly, most people prefer A1 since it has the same attributes as A2 with 1/6 the risk.
Now let’s run the numbers
Expected return of the new portfolio is the same whether we choose A1 or A2:
Volatility of the new portfolio if we choose A1:
Sharpe ratio of original portfolio = .33
Sharpe ratio when we add A1 = .049/.13363 or .3667
The Sharpe ratio improved by about 10%
Now what is the Sharpe ratio if we add A2 instead of A1.
First, we must compute the volatility. Go ahead, plug and chug…
That’s right, the volatility is the same!
The volatility of the new portfolio is the same whether we add A1 or A2 which means the new combined portfolio has the same improvement to Sharpe whether we add A1 or A2. This is true despite A2 having a far worse Sharpe than A1! It is counterintuitive because portfolio math and the role of correlation is not intuitive.
To see why, look at the formula for portfolio volatility:
Let’s zoom in on the last 2 terms which come from adding the second asset:
Plot of change in overall portfolio volatility vs volatility of prospective asset (A1 or A2)
As we increase the asset’s risk, the first term grows exponentially, and the second term shrinks linearly (remember, the correlation is negative). It turns out that, at least temporarily, the shrinking effect from the negative correlation outweighs the exponential term.
There are 2 observations to note once you are done reeling from the bizarre impact of correlation.
B1 | B2 | C1 | C2 | D1 | D2 | |
Return | 10.54% | 3.57% | 9.33% | 6.50% | 6.43% | -2.64% |
Volatility | 20.00% | 20.00% | 27.50% | 12.50% | 10.00% | 40.00% |
Correlation | .80 | -.20 | .40 | .40 | .50 | -.60 |
Most people agree:
The punchline, of course, is that every one of these assets improves the Sharpe of the portfolio by the same 10%. Your intuition would tell you would prefer a portfolio in the upper left green box since those assets have the best Sharpe (risk/reward), so it is probably uncomfortable to learn that the final portfolio is mathematically indifferent to all of these assets.
Correlation Is The Key
Here’s the same plot relating these equivalent portfolios by their respective correlations
As the correlation drops (corresponding to lines of “cooler” coloring), less return is required to deliver the same 10% improvement!
While Sharpe ratios are “mentally portable”, they are shockingly incomplete without being tied to correlation. To create a compact formula which links Sharpe ratios with correlation, it is helpful to view indifference curves.
RRRa = Sharp Ratio of prospective asset
RRRb = Sharp Ratio of original portfolio
If Relative RRR > 1 the Sharpe of the prospective asset is greater than the Sharpe ratio of the original portfolio
The indifference curve represents an equivalent tradeoff between Sharpe ratio and correlation for various mixing weights. For example, the light green line assumes you will allocate 20% of the original portfolio to the prospective asset.
Observations
The investor’s natural question when evaluating a new asset or investment is:
“What is required from an asset (in terms of return, risk and correlation) in order to add value to my portfolio?”
With math that can be verified in the paper’s appendix we find a very handy identity:
This equality describes what’s required, in an absolute bare-minimum mathematical sense, of a prospective asset in order to do no harm.
How to use it
For a given prospective Sharpe ratio, you very simply compute the maximum correlation the new asset can have to be accretive to the portfolio. For example, if the prospective asset has a Sharpe ratio of .10 and the original portfolio has a Sharp ratio of .40 then the prospective asset requires correlation no greater than .25 (ie .10/.40).
For a given correlation, you can compute the minimum required Sharpe ratio of the new asset to improve the portfolio. If the correlation is .80 and the original portfolio has a Sharpe ratio of .70 then the prospective asset must have a Sharpe ratio of at least .56 (ie .80 x .70).
Insights and Caveats
Breaking The Market’s outstanding post Optimal Portfolios For Two Assets
You will learn:
You can save your own copy here
You can also play with the numbers directly below
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